Sparsity of intersections with group subschemes within an abelian scheme

Abstract: I will talk about a unification of two bounded height results around abelian varieties. The first is Silverman’s specialization theorem, which states for an abelian scheme A/C with no fixed part over a curve C, that the set of points on C where the generic Mordell—Weil group fails to specialize injectively has bounded height. The second is by Habegger in an abelian variety: a suitable subvariety intersected with all torsion cosets up to complementary dimension gives a set of bounded height. I will take the point of view from unlikely intersections and discuss the key idea of the arithmetic part of the proof by homomorphism approximations using Ax—Schanuel results.

commutative algebraalgebraic geometrygroup theorynumber theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

Calgary Algebra and Number Theory Seminar

Series comments: This seminar series is partially supported by the Pacific Institute for the Mathematical Sciences (PIMS).